Repetition rate multiplication of femtosecond light pulses using a phase-locked all-pass fiber resonator

We describe an all-pass fiber resonator with active phase-locking capability for accurate multiplication of the repetition rate of femtosecond light pulses. The cavity length of the resonator is precisely controlled using the Pounder-Drever-Hall phase-locking technique so that the repetition rate is multiplied in stabilization to the Rb atomic clock. Our test result proves the proposed phase-locking scheme is an effective means of generating higher repetition rate pulses with no significant power loss while providing a high degree of long-term stability. ©2015 Optical Society of America OCIS codes: (140.7090) Ultrafast lasers; (140.4050) Mode-locked lasers; (140.3425) Laser stabilization; (120.3930) Metrological instrumentation. References and links 1. R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85(11), 2264–2267 (2000). 2. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrierenvelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000). 3. J. Ye, H. J. Kimble, and H. Katori, “Quantum state engineering and precision metrology using state-insensitive light traps,” Science 320(5884), 1734–1738 (2008). 4. J. J. McFerran, E. N. Ivanov, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, “Low noise synthesis of microwave signals from an optical source,” Electron. Lett. 41(11), 36–37 (2005). 5. A. Bartels, S. A. Diddams, C. W. Oates, G. Wilpers, J. C. Bergquist, W. H. Oskay, and L. Hollberg, “Femtosecond-laser-based synthesis of ultrastable microwave signals from optical frequency references,” Opt. Lett. 30(6), 667–669 (2005). 6. Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007). 7. S. T. Cundiff and A. M. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics 4(11), 760–766 (2010). 8. C.-H. Li, A. J. Benedick, P. Fendel, A. G. Glenday, F. X. Kärtner, D. F. Phillips, D. Sasselov, A. Szentgyorgyi, and R. L. Walsworth, “A laser frequency comb that enables radial velocity measurements with a precision of 1 cm,” Nature 452(7187), 610–612 (2008). 9. T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, L. Pasquini, A. Manescau, S. D’Odorico, M. T. Murphy, T. Kentischer, W. Schmidt, and T. Udem, “Laser frequency combs for astronomical observations,” Science 321(5894), 1335–1337 (2008). 10. G. C. Valley, “Photonic analog-to-digital converters,” Opt. Express 15(5), 1955–1982 (2007). 11. J. Kim, M. J. Park, M. H. Perrott, and F. X. Kärtner, “Photonic subsampling analog-to-digital conversion of microwave signals at 40-GHz with higher than 7-ENOB resolution,” Opt. Express 16(21), 16509–16515 (2008). 12. T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011). 13. K. Minoshima and H. Matsumoto, “High-accuracy measurement of 240-m distance in an optical tunnel by use of a compact femtosecond laser,” Appl. Opt. 39(30), 5512–5517 (2000). 14. J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. 29(10),


Introduction
Efforts are being made to bring the repetition rate of femtosecond light pulses upward, preferably to the upper GHz regime, in response to the need in diverse applications; optical frequency metrology [1][2][3], optical communications [4,5], arbitrary waveform generation [6,7], calibration of astronomical spectrographs [8,9], photonic analog-to-digital converters [10,11], microwave generation [12], and absolute distance measurements [13][14][15][16].The pulse repetition rate produced by an oscillator may be increased simply by shortening the cavity length of the oscillator.However, this approach has limitations as it reduces the pulse peak power which should be kept above a certain threshold to maintain the nonlinear mode locking to produce normal short pulses [17].The problem becomes more apparent for rare-earthdoped fiber lasers in which the gain medium has to be embedded along a lengthy fiber.The state-of-the-art repetition rate of rare-earth-doped fiber lasers demonstrates above 1 GHz, even though it is being limited by the rare-earth doping density of available gain fibers and its physical length [18,19].
Pulse-interleaving performed by incorporating extra devices at the exit of the oscillator permits producing higher repetition rate pulses.This approach of so-called repetition rate multiplication has been demonstrated so far using Fabry-Perot etalons [20,21], Mach-Zehnder interferometers [22,23] and sub-ring fiber resonators [24][25][26].These methods have their own merits and disadvantages particularly in terms of the hardware complexity and pulse energy loss.For example, a single Fabry-Perot cavity is enough to sort out of the optical modes of the frequency comb up to several tens of GHz spacing using the Hänsch-Couillaud locking technique [20], but the pulse energy is weakened in proportion to the filter spacing.In the case of unequal-arm Mach-Zehnder interferometers (MZIs), a single MZI leads to only doubling of the repetition rate, requiring multiple MZIs cascaded in series to achieve a high multiplication.Nonetheless, a 50% of the pulse energy is lost at every 2x2 coupler exit junction of each MZI.On the other hand, the fiber sub-ring resonator is most effective in conserving the pulse energy since it acts as an all-pass type resonator which transmits all the input pulse energy through the exit junction without significant loss [25].
In this investigation, we demonstrate a phase-locked sub-ring fiber resonator devised for stable multiplication of the repetition rate of an Er-doped fiber oscillator.The cavity length of the sub-ring resonator is controlled using the Pound-Drever-Hall (PDH) locking technique so that the temporal interval of pulse-interleaving is stabilized to the original pulses of the main Er-doped oscillator [27][28][29].Then the stability of multiplied pulses are tested to validate the proposed resonator with phase-locking capability as an effective means of generating higher repetition rate pulses with no significant power loss while providing a high degree of longterm stability.

All-pass fiber resonator for repetition rate multiplication
The fiber resonator intended to multiply the repetition rate in the time domain can be treated as a periodic spectral filter in the frequency domain, which suppresses the frequency modes of the frequency comb with an interval of the free spectral range (FSR) of the fiber resonator (Fig. 1(a)).In this study an all-pass fiber resonator was devised by combining a single-mode fiber with a fiber coupler of variable coupling ratio (Fig. 1(b)).The spectral transmittance T cavity of the fiber resonator is expressed as [27] where I in and I out are the input and output intensity of the all-pass fiber resonator.The other variables are; γ 0 the insertion loss, k r the resonant coupling ratio of the variable coupler, β the propagation constant and L the optical path length of the fiber resonator.As shown in Fig. 1(c), the spectral transmittance of the fiber resonator provides resonant peaks repeatedly at the positions where the condition of sin 2 (βL/2-π/4) = 1 is precisely met.For effective filtering, the transmittance at the resonant peaks is made null by tuning the coupling ratio so as to satisfy k r = (1-γ 0 )•exp(-2α 0 L) with α 0 being the net attenuation within the fiber resonator.Consequently, when the spectral modes of the main oscillator are overlapped with the transmittance of the resonator, the frequency modes located at zero-transmittance positions are eliminated.Figure 2 illustrates the optical layout of the phase-locked repetition rate multiplication system configured in this study.It is comprised of an Er-doped fiber oscillator, an all-pass fiber resonator and a Pound-Drever-Hall (PDH) phase-locking control unit.The Er-doped fiber femtosecond laser (Menlosystems, C-Fiber) was used as the main oscillator which is set to provide a pulse train of 100-fs duration at a 100-MHz repetition rate being stabilized to the Rb clock with a 10 −12 stability (10-s averaging) using the PLL technique [30,31].The output frequency of the main oscillator are then modulated using an electro-optic modulator (EOM, Newfocus, 4004) for subsequent PDH phase-locking control of the cavity length of the fiber resonator.The fiber resonator is comprised of a single-mode fiber (Corning, SMF-28) and a directional coupler with its coupling ratio being manually adjustable (Newport, F-CPL-1550).The coupling ratio of the resonator is first coarsely tuned so that the relative intensity fluctuation between the original and interleaved pulses is minimized by monitoring the multiplied pulses using an oscilloscope together with an RF spectrum analyzer.The middle point of the fiber resonator is separated and light-coupled via free space using two collimating lenses.One of the lenses is mounted on a translational stage coupled with a PZT actuator for fine motion control in order to precisely adjust the cavity length.The total optical path length (OPL) of the fiber resonator including the single-mode fiber and also the free space is controlled to be accurately half of the OPL of the main oscillator using the PDH technique, enabling the resonating modes of the resonator to overlap exactly with the original modes of the main oscillator.For the PDH locking, the EOM is set to modulate the optical phase of the main oscillator at 1 kHz with a 100 Hz modulation depth.The PDH error signal is then captured using a photo-detector and demodulated through a lock-in amplifier (Stanford Research Systems, SR510).The PDH error signal is loop-filtered and fed-back to a proportional-integral (PI) servo (Newport, LB1005) with a 1 kHz control bandwidth which controls the PZT actuator to lock the cavity length of the resonator to precisely half of the original oscillator cavity length.
More specifically, for the PDH locking control, the frequency ω(t) of the main oscillator is modulated by the EOM with a sinusoidal voltage input as [28,29] where σ is the modulation depth and Ω is the EOM modulation frequency.Assuming both σ and Ω are small, the time-varying output intensity of the fiber resonator is approximated as ( ) The derivative of the transmittance, i.e., dT cavity /dω, is then extracted by demodulation of the output intensity of Eq. ( 3) using the lock-in amplifier in synchronization with the modulation frequency Ω, and finally locked to the zero-crossing point as illustrated in Fig. 3.

Experiments and performance evaluation
Figure 4(a) shows a set of transmittance curves of the all-pass fiber resonator used in this study calculated with different values of the net attenuation and coupling-ratio so as to provide zero-transmittance at the resonant peaks.The cavity finesse decreases significantly from 60 to 3 while the net attenuation increases from −0.1 dB to −2.0 dB.Nevertheless, the linewidth of the resonant peaks is maintained within tens of MHz, which is narrow enough to spectrally filter 100 MHz-spaced spectral modes of the main oscillator to 200 MHz-spaced modes.In our experiments, the net-attenuation of the fiber cavity was selected to be −1.2 dB in consideration of the light-coupling loss occurring between the collimating lenses installed in the free space of the fiber resonator.Besides, based on the simulation results, the coupling ratio was set to ~55% in achieving the zero-transmittance at the resonant peaks.In order to characterize the transmittance curve of the fiber resonator and corresponding PDH phase-locking signal, a wavelength-tunable monochromatic distributed feedback (DFB) laser operating at a 1560 nm wavelength range was adopted as the input light source.While the input wavelength was detuned by an amount of 120 MHz, the transmittance intensity and also the PDH control signal were measured as shown in Fig. 4(b).The full-width-halfmaximum (FWHM) linewidth of the transmittance curve was measured ~35 MHz, which corresponds to the cavity finesse value of around 6.0.The PDH control signal was found to follow exactly the derivative of the transmittance curve with the slope around the zerocrossing point reaching 1.25 V/GHz at the center of the transmittance peak.The solid black line in Fig. 4(b) which was calculated with the coupling ratio and attenuation inside the fiber resonator using Eq. ( 1) showed a good agreement with the experimental transmittance curve (red circles).
Once the PDH phase-locking of the fiber resonator was done, the time and frequency domain characteristics were verified.First, the pulse train of two-fold multiplied pulses produced by the fiber resonator was monitored using an oscilloscope as shown in Fig. 5(a).Intensity fluctuation of ~7% shown in Fig. 5(a) is thought to be attributed to the free-running carrier-envelope offset frequency and also the frequency-jitter-to-intensity-noise conversion [20,21].The dispersion of the single-mode fiber inside the resonator plus other fiber sections was managed by adding a dispersion compensating fiber (Thorlabs, DCF38).The multiplied pulse duration was 267 fs when measured using a second-harmonic interferometric autocorrelator (Femtochrome, FR-103PD) as shown in Fig. 5(b), which was slightly broadened from the original duration of 100 fs of the main oscillator The output power was measured to 2.9 mW when a 3.3 mW input power was incident to the fiber cavity, which corresponds to a 1.29 dB loss.This power loss was mainly attributed to the coupling efficiency of the freespace optical delay line inside the fiber cavity.The multiplication result was also monitored using an RF-spectrum analyzer (RBW: 100 kHz, VBW: 100 kHz) as in Fig. 5(c) and 5(d), which shows the repetition-rate-doubled harmonics with suppression of unwanted modes by 40 dB in the RF domain.The relative amplitude decrease in the 200-MHz pulses was attributable to the limited detection bandwidth of the used photodetector (Menlosystems, FPD510).The long-term stability of the phase-locked multiplied pulses was monitored using a frequency counter (Pendulum, CNT91) synchronized to the Rb clock (SRS, FS727) over an interval of 2400 s in comparison to the 2f r original pulses of the main oscillator as plotted in Fig. 5(e), verifying no notable degradation of stability in the process of multiplication.The long-term locking stability for both cases shows 10 −12 at 10 s averaging, while the stability was ~10 −8 in the free-running state.With the aim of verifying the short-term characteristics of the repetition-rate-multiplied pulse train, a balance cross-correlator (BCC) permitting sub-femtosecond timing measurement was exploited [32][33][34][35].We used a type-II phase-matched second harmonic generation crystal which generates frequency doubled sub-pulses by providing two orthogonal polarization pulses [35].Balancing of the two sub-pulses' intensities of both the forward and backward propagation in the crystal which has polarization dependent GVD walk-off results in a S-shape curve as shown in the inset diagram of Fig. 6.The S-shape curve quantifies the temporal offset between the pulses under test and the reference pulses which enables precise measuring of timing jitter.As shown in Fig. 6, the optical path difference (OPD) between the two interferometer arms was set to 1.5 m, which corresponds to c/(2mf r ) with m = 1 and f r = 200 MHz.This OPD allows for interference between two sets of pulse trains -one is the original pulse train and the other the interleaved pulse train.The resulting BCC signal consequently represents the relative timing jitter between the original and interleaved pulses.In order to remove detritus timing jitters caused by other external noise sources, the BCC signal was fed back to the cavity-length-controlling PZT inside the main oscillator.This feedback loop locks the BCC signal to a zero voltage by controlling the pulse repetition rate by adjusting the intra-cavity PZT.This locking control effectively compensates for the timing jitter induced by the measurement itself not by the multiplication process.An alternative way is controlling one of the interferometer arm length using PZT [36].The inset diagram in Fig. 6 shows the s-shaped balanced cross-correlation signal at the given OPD of 1.5 m, indicating a high linearity of 2.84 mV/fs around the zero-crossing point with respect to the time difference between the reference and measurement pulses.In order to minimize the undesired timing jitter originated from the BCC control, the locking bandwidth of the BCC control should be set to be as low as possible [34].Thereby, the BCC control works to compensate the environmental change but generate insignificant level timing jitter.In our experiment, the BCC control loop was set to have a 30 Hz-bandwidth by using a PI servo controller (Newport, LB1005).A time trace of the balanced cross-correlation signal under the locked status was recorded with a 4 kHz update rate as shown in Fig. 7(a).The standard deviation of the timing jitter was evaluated 1.65 fs.Furthermore, a frequency domain analysis was performed as shown in Fig. 7(b) by Fourier-transforming the time trace of the jitter signal.There was found no distinct noise peaks including the EOM modulation from 1 Hz to 2 kHz due to the high PDH feedback bandwidth.The integrated timing jitter over the bandwidth was 1.61 fs which implies that there was no significant degradation in the timing jitter during the repetition rate multiplication process.

Conclusions
We demonstrated a stable method of repetition-rate multiplication of femtosecond light pulses by PDH phase-locking of an all-pass fiber resonator to the harmonics of the original pulses.Long-term and also short-term stability tests performed on the doubled pulse repetition rate revealed no significant degradation in not only the frequency stability and but also the timing jitter.Our test result proves the proposed phase-locking scheme is an effective means of generating higher repetition rate pulses with no significant power loss while providing a high degree of stability.Higher-order repetition-rate multiplication to several GHz could be realized by shortening the cavity length or using the cascaded cavity configuration.

Fig. 1 .
Fig. 1.Multiplication of the pulse repetition rate f r by an optical fiber resonator.(a) Conceptual diagram of f r multiplication in the frequency domain.(b) Schematic of all-pass fiber resonator structure for f r multiplication.(c) Selective filtering of optical modes by the transmittance of the fiber resonator (solid black line).Red solid lines and red dotted line refer to the survived and filtered optical modes, respectively.

Fig. 3 .
Fig. 3. Conceptual diagram of the PDH lock-in signal, (a) Variation of the output amplitude (red) with respect to the frequency mismatch between the frequency modulated input (green) and transmittance peak (black).The point 'C' corresponds to the zero-crossing PDH signal of the steepest slope, while 'B' and 'D' refer to the maximum and minimum amplitudes.(b) Sshape curve for the PDH lock-in technique.

Fig. 4 .
Fig. 4. (a) (Simulation) Resonant transmittance curves of the fiber resonator with different values of net cavity attenuation and coupling ratio.(b) (Experiment) Transmittance curve (red circle) and corresponding PDH locking signal (blue square) measured by detuning the wavelength of a monochromatic DFB laser.Black solid line refers to the transmittance curve fitted with −1.2 dB attenuation and 55% coupling-ratio.

Fig. 5 .
Fig. 5. Experimental results of f r multiplication.(a) Multiplied femtosecond pulses in the time domain.(b) Temporal pulse duration of multiplied pulses.(c) Original RF spectrum of the main oscillator with a 100 MHz pulse repetition rate, (d) RF spectrum of two-fold multiplied pulses of a 200 MHz repetition rate.(e) Frequency stability of the second harmonic of the original pulses (blue dots) locked to the Rb reference clock and two-fold multiplied pulses (red dots).

Fig. 7 .
Fig. 7. Relative timing jitter evaluation.(a) Time trace of timing jitter recorded using a 4 kHz sampling oscilloscope.(b) Timing jitter spectral density by Fourier transform of time trace signal (red) and integrated timing jitter from 1 Hz to 2 kHz (black).